We will now briefly survey some of the potential applications for the ZH-calculus. We begin with the simple observation that \(n\) -ary H-boxes let us generalise the usual string diagrammatic description the controlled-Z gate (as in e.g. the ZX calculus) to an \(n\) -controlled-Z gate:
n-controlled-Z
Using the decomposition of controlled-Z gates above, a representation of graph states as ZX-diagrams was given in [1]}, which in turn gave a fully diagrammatic derivation of the local complementation law for graph states [1]} and a new procedure for extracting circuits from certain computations in the one-way model [3]}. Passing from \(\wedge Z\)  to \(\wedge ^n Z\)  gives an analogous representation for hypergraph states:
\( {gs-graph-s}\qquad \textrm {\Large \) 
